Error: Incorrect local algebra for Q(zeta3)
Defining polynomial
\(x^{10} + 17\)
|
Invariants
Base field: | $\Q_{17}$ |
Degree $d$: | $10$ |
Ramification index $e$: | $10$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $9$ |
Discriminant root field: | $\Q_{17}(\sqrt{17})$ |
Root number: | $-1$ |
$\Aut(K/\Q_{17})$: | $C_2$ |
This field is not Galois over $\Q_{17}.$ | |
Visible Artin slopes: | $[\ ]$ |
Visible Swan slopes: | $[]$ |
Means: | $\langle\ \rangle$ |
Rams: | $(\ )$ |
Jump set: | undefined |
Roots of unity: | $16 = (17 - 1)$ |
Intermediate fields
$\Q_{17}(\sqrt{17})$, 17.1.5.4a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
Unramified subfield: | $\Q_{17}$ |
Relative Eisenstein polynomial: |
\( x^{10} + 17 \)
|
Ramification polygon
Residual polynomials: | $z^{9} + 10 z^{8} + 11 z^{7} + z^{6} + 6 z^{5} + 14 z^{4} + 6 z^{3} + z^{2} + 11 z + 10$ |
Associated inertia: | $4$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois degree: | $40$ |
Galois group: | $C_2\times F_5$ (as 10T5) |
Inertia group: | $C_{10}$ (as 10T1) |
Wild inertia group: | $C_1$ |
Galois unramified degree: | $4$ |
Galois tame degree: | $10$ |
Galois Artin slopes: | $[\ ]$ |
Galois Swan slopes: | $[]$ |
Galois mean slope: | $0.9$ |
Galois splitting model: | $x^{10} - 17$ |